Compute Karl Pearson's coefficient of correlation in the following series relating to cost of living and wages.

Steps to Solve

1. List the Data
   We have two sets of data:
   Wages: $[100, 101, 102, 100, 98, 97, 98, 96, 95]$
   Cost of Living: $[98, 99, 99, 97, 95, 95, 94, 90, 91]$

2. Calculate Averages
   Calculate the mean for each dataset.
   Mean of Wages ($\bar{X}$):
   $$ \bar{X} = \frac{100 + 101 + 102 + 100 + 98 + 97 + 98 + 96 + 95}{9} = \frac{987}{9} = 109.67 $$

Mean of Cost of Living ($\bar{Y}$):
$$ \bar{Y} = \frac{98 + 99 + 99 + 97 + 95 + 95 + 94 + 90 + 91}{9} = \frac{ 88.44}{9} = 97.56 $$

3. Calculate Deviations from the Mean
   For each data point, subtract the mean from the point:
   $$ D_X = [100 - 99.67, 101 - 99.67, 102 - 99.67, 100 - 99.67, 98 - 99.67, 97 - 99.67, 98 - 99.67, 96 - 99.67, 95 - 99.67] $$
   $$ D_Y = [98 - 97.56, 99 - 97.56, 99 - 97.56, 97 - 97.56, 95 - 97.56, 95 - 97.56, 94 - 97.56, 90 - 97.56, 91 - 97.56] $$

4. Calculate Products of Deviations
   Now calculate $D_X \cdot D_Y$ for pairs:
   $$ D_X \cdot D_Y = [(100 - 99.67)(98 - 97.56), (101 - 99.67)(99 - 97.56), ...] $$

5. Sum of Squares
   Calculate the sum of squares for Wages and Cost of Living:
   $$ SS_X = \sum (D_X^2) $$
   $$ SS_Y = \sum (D_Y^2) $$

6. Compute Pearson’s Correlation Coefficient
   Use the formula for Pearson's correlation coefficient:
   $$ r = \frac{\sum (D_X \cdot D_Y)}{\sqrt{SS_X \cdot SS_Y}} $$